Traverse adjustment
Metes and bounds deed traverse: closure, precision and area
A five-call deed description is reduced to a traverse. It closes at 1:4,958, which is examined against what a deed of that vintage should be expected to deliver, then adjusted and its area compared with the recited acreage.
Given
- A deed describes a small parcel by metes and bounds, beginning at an iron pin at the point of beginning and running as follows.
- First call: thence N 12°30′00″ E, 210.50 feet to an iron pin.
- Second call: thence N 78°15′00″ E, 165.20 feet to an iron pin.
- Third call: thence S 14°10′00″ E, 198.60 feet to an iron pin.
- Fourth call: thence S 80°05′00″ W, 172.40 feet to an iron pin.
- Fifth call: thence S 79°00′00″ W, 87.75 feet to the point of beginning.
- The deed recites that the parcel contains 0.95 acre more or less.
- The point of beginning is assigned N 1000.00, E 1000.00 on an assumed grid.
- Distances to 0.01 ft; bearings as recited, to the whole minute.
Required
- Latitudes and departures of the five calls, and the closure of the description.
- The precision of the deed as written.
- Compass-rule adjusted coordinates of the five corners.
- The area, and a comparison with the recited acreage.
Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.
Worked solution
Reduce the calls to azimuths
A deed states directions as quadrant bearings, which is how they were observed and how they should be recited. Computation is easier in azimuths, so convert first, using the four standard rules: a NE bearing is the azimuth, a SE bearing is 180 degrees less the bearing, a SW bearing is 180 plus, and a NW bearing is 360 less.
Nothing is lost in the conversion and the bearings should be carried alongside so that the computation sheet can be read against the deed line by line.
N 12°30′00″ E → azimuth 12°30′00″
N 78°15′00″ E → azimuth 78°15′00″
S 14°10′00″ E → azimuth 180° − 14°10′00″ = 165°50′00″
S 80°05′00″ W → azimuth 180° + 80°05′00″ = 260°05′00″
S 79°00′00″ W → azimuth 180° + 79°00′00″ = 259°00′00″Latitudes, departures and the closure
Resolve each call. The sums are the error of closure of the description as written, and they are the first thing a retracement surveyor computes before setting foot on the ground.
A description that does not close at all, by tens of feet, usually signals a missing or garbled call. A description that closes to a few tenths, as this one does, is a normal deed and the closure is telling you about the quality of the original survey, not about a transcription error.
| Call | Bearing | Azimuth | Distance (ft) | Latitude (ft) | Departure (ft) |
|---|---|---|---|---|---|
| 1 | N 12°30′00″ E | 12°30′00″ | 210.50 | 205.510 | 45.561 |
| 2 | N 78°15′00″ E | 78°15′00″ | 165.20 | 33.642 | 161.738 |
| 3 | S 14°10′00″ E | 165°50′00″ | 198.60 | −192.560 | 48.606 |
| 4 | S 80°05′00″ W | 260°05′00″ | 172.40 | −29.690 | −169.824 |
| 5 | S 79°00′00″ W | 259°00′00″ | 87.75 | −16.743 | −86.138 |
| Sums | 834.45 | 0.158 | −0.057 |
Precision of the description
The linear misclosure is the resultant of the two sums; the precision is the perimeter divided by it.
At 1:4,958 this deed is unremarkable. Bearings recited only to the whole minute cannot do much better: one minute of arc on a 200 ft call is 0.06 ft, and five such calls with independent rounding will typically produce a tenth or two of closure all by themselves. Judging an older deed against a modern 1:10,000 traverse specification misreads what the numbers are.
What matters for retracement is whether the closure is consistent with the recited precision of the calls, and whether the misclosure has any direction that points at a particular call. Here it does not: 160 degrees is not close to any call bearing or its reverse.
e = √(0.1583² + 0.0571²) = √(0.025059 + 0.003260) = 0.1683 ft
precision = 834.45 / 0.1683 = 4,958, that is 1:4,958
azimuth of the closure line = 160°09′
expected closure from bearings rounded to the whole minute: about 0.06 ft per 200 ft callAdjust and compute coordinates
Adjust by the compass rule so the figure closes, then accumulate coordinates from the point of beginning.
This adjustment is a computational convenience for finding the area and for plotting. It is not a statement about where any monument is. In retracement the found monuments control, and the adjusted coordinates are only a first approximation to be checked against them.
| Call | Lat correction (ft) | Dep correction (ft) | Adjusted latitude (ft) | Adjusted departure (ft) | To corner | Northing (ft) | Easting (ft) |
|---|---|---|---|---|---|---|---|
| 1 | −0.040 | +0.014 | 205.470 | 45.575 | 2 | 1205.47 | 1045.57 |
| 2 | −0.031 | +0.011 | 33.610 | 161.750 | 3 | 1239.08 | 1207.32 |
| 3 | −0.038 | +0.014 | −192.598 | 48.620 | 4 | 1046.48 | 1255.94 |
| 4 | −0.033 | +0.012 | −29.723 | −169.812 | 5 | 1016.76 | 1086.13 |
| 5 | −0.017 | +0.006 | −16.760 | −86.132 | POB | 1000.00 | 1000.00 |
Area and the recited acreage
Compute the area from the adjusted coordinates by the shoelace sum, then convert at 43,560 square feet to the acre exactly.
The computed 0.952 acre agrees with the recited 0.95 acre more or less. That agreement is worth having: a computed area that differs materially from the recitation is a warning that a call has been misread, that the description omits a curve, or that the parcel described is not the parcel intended.
2A = Σ (Eᵢ Nᵢ₊₁ − Eᵢ₊₁ Nᵢ) = 82,971.23 ft²
A = 41,485.62 ft²
A = 41,485.62 / 43,560 = 0.952 acre
recited: 0.95 acre more or lessAnswer
- Latitude closure +0.158 ft, departure closure −0.057 ft, linear misclosure 0.168 ft over a 834.45 ft perimeter.
- Precision of the description as written: 1:4,958, which is normal for bearings recited to the whole minute.
- Adjusted coordinates from a point of beginning at 1000.00 / 1000.00: corner 2 at 1205.47 / 1045.57; corner 3 at 1239.08 / 1207.32; corner 4 at 1046.48 / 1255.94; corner 5 at 1016.76 / 1086.13.
- Area 41,485.62 square feet, or 0.952 acre, against the recited 0.95 acre more or less.
- The description is internally consistent; nothing in the closure points at a garbled call.
Check
The adjusted latitude and departure columns sum to zero, and the fifth call returns to the point of beginning at 1000.00 / 1000.00 exactly.
Compute the area a second way, by double meridian distances from the same adjusted latitudes and departures. The DMD column sums to 82,971.23 square feet, twice the area, matching the coordinate method. A disagreement between the two would mean the corners had been taken out of boundary order.
Check the closure against what the recited precision can deliver. Five calls with bearings rounded to the whole minute, on distances averaging 167 ft, each carry up to about 0.05 ft of direction rounding; combined in quadrature that is roughly 0.11 ft, of the same order as the observed 0.17 ft. The description is consistent with itself.
Independent area sanity check: the parcel spans 239 ft north to south and 256 ft east to west, bounding it at 1.40 acres, and it is visibly a quadrilateral with one corner cut, so 0.9 to 1.1 acres is expected. The computed 0.952 acre and the recited 0.95 acre both fall there.
More traverse adjustment
All in this category- Closed-loop traverse from field angles to compass-rule coordinatesA five-sided loop traverse is taken from raw interior angles and taped distances through angular closure, azimuths, latitudes and departures, misclosure and precision, to compass-rule adjusted coordinates.
- Transit-rule adjustment compared with the compass ruleThe same five-sided loop is readjusted by the transit rule, which distributes the closure in proportion to each latitude and departure rather than to course length, and the two sets of coordinates are compared.
- Closed-link traverse between two pairs of control monumentsA four-course traverse runs from one control pair to another. Azimuth closure comes from the two published control azimuths and position closure from the published coordinates of the far monument.